\( over a bounded subset \)\( is denoted by \) [f] := 1||f   dx\(.See for instance \cite {grafakos,Stein93} for useful properties of the BMO space. In what follows, we denote the unit ball by \) B\( and the ball with radius \) R\( centred at the origin by \) B(R)\(.\) We say that a vector field \(v\)  belongs to the space \(BMO^{-1}(\mathbb {R}^2)\) , if there exists a matrix \(F\in BMO(\mathbb {R}^2;\mathbb {R}^{2 \times 2})\)  such that \(v = \operatornamewithlimits{\mbox{div}}F\) , and this is the only property of this function space we need in this work; see for instance [1]} for discussions regarding this space.
